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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions
17.23 CHALLENGE 2σ control charts. Some special situations call for 2σ control charts. That is, the control limits for a statistic Q will be μQ ± 2σQ. Suppose that you know the process mean μ and standard deviation σ and will plot x and s from samples of size n.(a) What are the 2σ control
17.22 CHALLENGE Alternative control limits. American and Japanese practice uses 3σ control charts.That is, the control limits are three standard deviations on either side of the mean. When the statistic being plotted has a Normal distribution, the probability of a point outside the limits is about
17.21 CHALLENGE Determining the probability of detection.An x chart plots the means of samples of size 4 against center line CL = 700 and control limits LCL = 685 and UCL = 715. The process has been in control.(a) What are the process mean and standard deviation?(b) The process is disrupted in a
17.20 Identifying special causes on control charts.The process described in Exercise 17.18 goes out of control. Investigation finds that a new type of yarn was recently introduced. The pH in the kettles is influenced by both the dye liquor and the yarn.Moreover, on a few occasions a faulty valve on
17.19 Control charts for a mounting-hole process.Figure 17.10 reproduces a data sheet from the floor of a factory that makes electrical meters.5 The sheet shows measurements of the distance between two mounting holes for 18 samples of size 5. The heading informs us that the measurements are in
17.18 Control limits for a dyeing process. The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color.There are 5 kettles, all of which
17.17 Control limits for a milling process. The width of a slot cut by a milling machine is important to the proper functioning of a hydraulic system for large tractors. The manufacturer checks the control of the milling process by measuring a sample of 5 consecutive items during each hour’s
17.16 More on the tablet compression process.Exercise 17.15 concerns process control data on the hardness of tablets for a pharmaceutical product. Table 17.5 gives data for 20 new samples of size 4, with the x and s for each sample. The process has been in control with mean at the target value μ =
17.15 Control charts for a tablet compression process.A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets is measured in order to control the compression process.
17.14 CHALLENGE Causes of variation in the time to respond to an application. The personnel department of a large company records a number of performance measures. Among them is the time required to respond to an application for employment, measured from the time the application arrives. Suggest
17.13 Control limits for a meat-packaging process.A meat-packaging company produces 1-pound packages of ground beef by having a machine slice a long circular cylinder of ground beef as it passes through the machine. The timing between consecutive cuts will alter the weight of each section. Table
17.12 Control limits for air conditioner thermostats. A maker of auto air conditioners checks a sample of 4 thermostatic controls from each hour’s production.The thermostats are set at 75◦F and then placed in a chamber where the temperature is raised gradually.The temperature at which the
17.11 Making a Pareto chart. Continue the study of the process of getting to work or class on time(Exercise 17.7). If you kept good records, you could make a Pareto chart of the reasons (special causes)for late arrivals at work or class. Make a Pareto chart that you think roughly describes your own
17.10 Constructing another Pareto chart. A large hospital finds that it is losing money on surgery due to inadequate reimbursement by insurance companies and government programs. An initial study looks at losses broken down by diagnosis.Government standards place cases into Diagnostic Related
17.9 Constructing a Pareto chart. Comparisons are easier if you order the bars in a bar graph by height.A bar graph ordered from tallest to shortest bar is sometimes called a Pareto chart, after the Italian economist who recommended this procedure.Pareto charts are often used in quality studies to
17.8 Determining sources of common and special cause variation. Refer to the previous exercise. The time at which you reach work or class varies from day to day, and your planning must allow for this variation. List several common causes of variation in your arrival time. Then list several special
17.7 Constructing a flowchart. Each weekday morning, you must get to work or to your first class on time. Make a flowchart of your daily process for doing this, starting when you wake. Be sure to include the time at which you plan to start each step.
17.6 Changing the sample size n. Refer to Exercise 17.5. What happens to the center line and control limits if(a) the owner samples 4 consecutive sandwich orders?(b) the owner samples 8 consecutive sandwich orders?(c) the owner uses minutes rather than seconds as the units?
17.5 An x control chart for sandwich orders. A sandwich shop owner takes a daily sample of 6 consecutive sandwich orders at random times during the lunch rush and records the time it takes to complete each order. Past experience indicates that the process mean should be μ = 168 seconds and the
17.4 Examples of special cause variation in bicycling times. Jeannine participates in bicycle road races. She regularly rides 25 kilometers over the same course in training. Her time varies a bit from day to day but is generally stable. Give several examples of special causes that might raise
17.3 Considering common and special cause variation. In Exercise 17.1, you described a process that you know well. What are some sources of common cause variation in this process? What are some special causes that might, at times, drive the process out of control?
17.2 What variables to measure? Based on your description of the process in Exercise 17.1, suggest specific variables that you might measure in order to(a) assess the overall quality of the process.(b) gather information on a key step within the process.
17.1 Describing your process. Choose a process that you know well, preferably from a job you have held. If you lack experience with actual business processes, choose a personal process such as cooking scrambled eggs or brushing your teeth. Make a flowchart of the process.Make a cause-and-effect
16.87 More on the assessment of a citizen-police program. The previous exercise applied significance tests to the Hyde Park burglary data. We might also apply confidence intervals.(a) Bootstrap the difference in mean monthly burglary counts. Make a histogram and a Normal quantile plot of the
16.86 Assessment of a citizen-police program. The following table gives the number of burglaries per month in the Hyde Park neighborhood of Chicago for a period before and after the commencement of a citizen-police program(a) Plot both sets of data. Are the distributions skewed or roughly
16.85 More on the readability of advertisements.The researchers in the study described in the previous exercise expected higher word counts in magazines aimed at people with high education level. Do a permutation test to see if the data support this expectation. State hypotheses, give a P-value,
16.84 Readability of magazine advertisements.Is there a difference in the readability of advertisements in magazines aimed at people with varying educational levels? Here are word counts in randomly selected ads from magazines aimed at people with high and middle education levels:21 Education level
16.83 CHALLENGE Prevalence of frequent binge drinking.Examples 8.9 (page 507) and 8.11 (page 513) examine survey data on binge drinking among college students. Here are data on the prevalence of frequent binge drinking among female and male students:20 Sample Binge Gender size drinkers Men 5,348
16.82 More on nurses’ use of latex gloves. In the previous exercise, you did a one-sided permutation test to compare means before and after an intervention. If you are mainly interested in whether or not the effect of the intervention is significant at the 5% level, an alternative approach is to
16.81 Nurses’ use of latex gloves. Nurses in an innercity hospital were unknowingly observed on their use of latex gloves during procedures for which glove use is recommended.19 The nurses then attended a presentation on the importance of glove use. One month after the presentation, the same
16.80 The standard deviation of returns on an investment. In financial theory, the standard deviation of returns on an investment is used to describe the risk of the investment. The idea is that an investment whose returns are stable over time is less risky than one whose returns vary a lot. The
16.79 Comparing 2001 and 2002 real estate prices.We have compared the selling prices of Seattle real estate in 2002 (Table 16.1) and 2001 (Table 16.5). Let’s compare 2001 and 2000. Here are the prices (thousands of dollars) for 20 random sales in Seattle in the year 2000:333 126.5 207.5 199.5
16.78 CHALLENGE More on social distress and brain activity. Use the bootstrap to obtain a 95% confidence interval for the correlation in the population of all similar subjects from the data in the previous exercise.(a) The permutation distribution in the previous exercise was reasonably Normal,
16.77 Social distress and brain activity. Exercise 2.17(page 97) describes a study that suggests that the“pain” caused by social rejection really is pain, in the sense that it causes activity in brain areas known to be activated by physical pain. Here are data for 13 subjects on degree of
16.76 Are female personal trainers, on average, younger? A fitness center employs 20 personal trainers. Here are the ages in years of the female and male personal trainers working at this center:Male 25 26 23 32 35 29 30 28 31 32 29 Female 21 23 22 23 20 29 24 19 22 (a) Make a back-to-back
16.75 Bootstrap confidence interval for the median.Your software can generate random numbers that have the uniform distribution on 0 to 1. Figure 4.9(page 264) shows the density curve. Generate a sample of 50 observations from this distribution.(a) What is the population median? Bootstrap the
16.74 More on average CO2 emissions. In Exercise 16.5 (page 16-12), you constructed the bootstrap distribution for the average carbon dioxide (CO2)emissions. Re-create this distribution here.(a) Based on the distribution, do you expect a bootstrap t confidence interval to be reasonable?Explain.(b)
16.73 More bootstrap confidence intervals of the trimmed mean. The bootstrap distribution of the 25% trimmed mean for the Seattle real estate sales (Figure 16.7) is not strongly skewed. We were therefore willing in Examples 16.5 and 16.8 to use the bootstrap t and bootstrap percentile confidence
16.72 CHALLENGE Comparing two operators. Exercise 7.39 (page 445) gives these data on a delicate measurement of total body bone mineral content made by two operators on the same 8 subjects:16 Subject Operator 1 2 3 4 5 6 7 8 1 1.328 1.342 1.075 1.228 0.939 1.004 1.178 1.286 2 1.323 1.322 1.073
16.71 A bootstrap comparison of variances. Are the variances of decreases in blood pressure equal in populations of black men given calcium and given a placebo? Example 7.22 (page 475) applied the F test for equality of variances to the data in Table 16.7. This test is unreliable because it is
16.70 More on calcium intake and blood pressure.We prefer measured data to the success/failure data given in Exercise 16.68. Table 16.7 gives the actual values of seated systolic blood pressure for this experiment. Example 7.20 (page 463)applies the pooled t test (a procedure that we do not
16.69 Bootstrap confidence interval for the difference in proportions.We want a 95% confidence interval for the difference in the proportions of reduced blood pressure between a population of black men given calcium and a similar population given a placebo. Summary data appear in Exercise 16.68.(a)
16.68 Calcium intake and blood pressure. Does added calcium intake reduce the blood pressure of black men? In a randomized comparative double-blind trial, 10 men were given a calcium supplement for twelve weeks and 11 others received a placebo.Whether or not blood pressure dropped was recorded for
16.67 Permutation test for the ratio of standard deviations. In Exercise 16.55 we compared the mean repair times for Verizon (ILEC) and CLEC customers. We might also wish to compare the variability of repair times. For the data in Exercise 16.55, the F statistic for comparing sample variances is
16.66 CHALLENG E Bootstrap confidence interval for the ratio. Here is one conclusion from the data in Table 16.6, described in Exercise 16.64: “The mean serum retinol level in uninfected children was 1.255 times the mean level in the infected children. A 95% confidence interval for the ratio of
16.65 Methods of resampling. In Exercise 16.64, we did a permutation test for the hypothesis“no difference between infected and uninfected children” using the ratio of mean serum retinol levels to measure “difference.” We might also want a bootstrap confidence interval for the ratio of
16.64 Comparing serum retinol levels. The formal medical term for vitamin A in the blood is serum retinol. Serum retinol has various beneficial effects, such as protecting against fractures. Medical researchers working with children in Papua New Guinea asked whether recent infections reduce the
16.63 Comparing average tree diameters. In Exercise 7.105 (page 480), the standard deviations of the tree diameter for the northern and southern regions of the tract were compared. The test is unreliable because it is sensitive to non-Normality of the data.Perform a permutation test using the F
16.62 Testing the correlation between salaries and batting averages. Table 16.2 contains the salaries and batting averages of a random sample of 50 Major League Baseball players. Can we conclude that the correlation between these variables is greater than 0 in the population of all players?(a)
16.61 Testing the correlation between Treasury bill and stock returns. In Exercise 16.45, we assessed the significance of the correlation between returns on Treasury bills and common stocks by creating bootstrap confidence intervals. If a 95% confidence interval does not cover 0, the observed
16.60 Comparing mpg calculations. Exercise 7.35(page 444) gives data on a comparison of driver and computer mpg calculations. This is a matched pairs study, with mpg values for 20 fill-ups. We conjecture that the computer is giving higher mpg values.(a) Carry out the matched pairs t test. That is,
16.59 Assessment of a summer language institute.Exercise 7.41 (page 446) gives data on a study of the effect of a summer language institute on the ability of high school language teachers to understand spoken French. This is a matched pairs study, with scores for 20 teachers at the beginning
16.58 Comparing median sales prices. Because distributions of real estate prices are typically strongly skewed, we often prefer the median to the mean as a measure of center. We would like to test the null hypothesis that Seattle real estate sales prices in 2001 and 2002 have equal medians.Sample
16.57 CHALLENGE When is a permutation test valid?You want to test the equality of the means of two populations. Sketch density curves for two populations for which(a) a permutation test is valid but a t test is not.(b) both permutation and t tests are valid.(c) a t test is valid but a permutation
16.56 Standard deviation of the estimated P-value.The estimated P-value for the DRP study (Example 16.12) based on 999 resamples is P = 0.015.For the Verizon study (Example 16.13) the estimated P-value for the test based on x1 − x2 is P = 0.0183 based on 500,000 resamples. What is the approximate
16.55 Comparing repair times in hours. Verizon uses permutation testing for hundreds of comparisons, such as between different time periods, between different locations, and so on. Here is a sample from another Verizon data set, containing repair times in hours for Verizon (ILEC) and CLEC
16.54 Permutation test of real estate prices. Table 16.1 contains the selling prices for a random sample of 50 Seattle real estate transactions in 2002. Table 16.5 contains a similar random sample of sales in 2001. Test whether the means of the two random samples of the 2001 and 2002 real estate
16.53 A small-sample permutation test. To illustrate the process, let’s perform a permutation test by hand for a small random subset of the DRP data(Example 16.12). Here are the data:Treatment group 57 53 Control group 19 37 41 42(a) Calculate the difference in means xtreatment − xcontrol
16.52 Declaring significance. Suppose a one-sided permutation test based on 200 permutation resamples resulted in a P-value of 0.04. What is the approximate standard deviation of this value? Would you feel comfortable declaring the result significant at the 5% level? Explain.
16.51 Is use of a permutation test valid? Suppose a professor wants to compare the effectiveness of two different instruction methods. By design, one method is more team oriented, and so he expects the vari ability in individual tests scores for this method to be smaller. Is a permutation test to
16.50 The effect of outliers. We know that outliers can strongly influence statistics such as the mean and the least-squares line. Example 7.7 (page 428)describes a matched pairs study of disruptive behavior by dementia patients. The differences in Table 7.2 show several low values that may be
16.49 CHALLENGE Predicting stock returns. Continue your study of historical returns on Treasury bills and common stocks, begun in Exercise 16.45, by regressing stock returns on T-bill returns.(a) Request a plot of the residuals against the explanatory variable and a Normal quantile plot of the
16.48 CHALLENGE Predicting field measurements. Continue your study of field measurements versus laboratory measurements of defects in the Trans-Alaska Oil Pipeline, begun in Exercise 16.44, by regressing field measurement result on laboratory measurement result.(a) Request a plot of the residuals
16.47 Predicting salary. Table 16.2 gives data on a sample of 50 baseball players.(a) Find the least-squares regression line for predicting salary from batting average.(b) Bootstrap the regression line and give a 95%confidence interval for the slope of the population regression line.(c) In the
16.46 CHALLENGE Bootstrap distribution for the slope β1.Describe carefully how to resample from data on an explanatory variable x and a response variable y to create a bootstrap distribution for the slope b1 of the least-squares regression line.(Software such as S-PLUS automates resampling methods
16.45 The correlation between Treasury bills and common stock returns. Figure 2.7 (page 96)shows a very weak relationship between returns on Treasury bills and returns on common stocks.The correlation is r = −0.113. We wonder if this is significantly different from 0. To find out, bootstrap the
16.44 The correlation between field and lab measurements. Figure 2.3 (page 90) is a scatterplot of field versus laboratory measurements of the depths of 100 defects in the Trans-Alaska Oil Pipeline. The correlation is r = 0.944. Bootstrap the correlation for these data. (The data are in the file
16.43 Bootstrap confidence intervals for the difference in average repair times. Example 16.6 (page 16-17) considers the mean difference between repair times for Verizon (ILEC) customers and customers of competing carriers (CLECs). The bootstrap distribution is non-Normal with strong left-skewness,
16.42 CHALLENGE Bootstrap confidence interval for the CLEC data. The CLEC data for Example 16.6 are strongly skewed to the right. The 23 CLEC repair times appear in Exercise 16.26 (page 16-23).(a) Bootstrap the mean of the data. Based on the bootstrap distribution, which bootstrap confidence
16.41 CHALLENGE The effect of decreasing sample size.Exercise 16.11 (page 16-12) gives an SRS of 20 of the 72 guinea pig survival times in Table 1.8.The bootstrap distribution of x from this sample is clearly right-skewed. Give a 95% confidence interval for the population mean μ based on these
16.40 Bootstrap confidence intervals for the standard deviation s. We would like a 95% confidence interval for the standard deviation σ of Seattle real estate prices. Your work in Exercise 16.19 probably suggests that it is risky to bootstrap the sample standard deviation s from the sample in
16.39 Bootstrap confidence intervals for the average survival time. The distribution of the 72 guinea pig survival times in Table 1.8 (page 29) is strongly skewed. In Exercise 16.17 (page 16-22) you found a bootstrap t confidence interval for the population mean μ, even though some skewness
16.38 Bootstrap confidence intervals for the average audio file length. In Exercise 16.13, you found a bootstrap t confidence interval for the population mean μ. Careful examination of the bootstrap distribution reveals a slight skewness in the right tail. Is this something to be concerned
16.37 BCa and tilting intervals for the correlation coefficient. Find the BCa and tilting 95%confidence intervals for the correlation between baseball salaries and batting averages, from the data in Table 16.2. Are these more accurate intervals in general agreement with the bootstrap t and
16.36 More on using bootstrapping to check traditional methods. Continue to work with the data given in Exercise 16.34.(a) Find the bootstrap BCa or tilting 95%confidence interval.(b) Does your opinion of the robustness of the one-sample t confidence interval change when comparing it to the BCa or
16.35 Comparing bootstrap confidence intervals. The graphs in Figure 16.15 do not appear to show any important skewness in the bootstrap distribution of the correlation for Example 16.9. Compare the bootstrap percentile and bootstrap t intervals for the correlation, given in the discussion of
16.34 Using bootstrapping to check traditional methods. Bootstrapping is a good way to check if traditional inference methods are accurate for a given sample. Consider the following data:108 107 113 104 94 100 107 98 112 97 98 95 95 97 99 95 97 90 109 102 89 101 93 95 105 91 96 104 95 87 91 101 119
16.33 Confidence interval for the Normal data set.In Exercise 16.21 (page 16-22) you bootstrapped the mean of a simulated SRS from the standard Normal distribution N(0, 1) and found the standard t and bootstrap t 95% confidence intervals for the mean.(a) Find the bootstrap percentile 95% confidence
16.32 Confidence interval for the average IQ score.The distribution of the 60 IQ test scores in Table 1.3 (page 13) is roughly Normal (see Figure 1.7), and the sample size is large enough that we expect a Normal sampling distribution. We will compare confidence intervals for the population mean IQ
16.31 Bootstrap percentile confidence interval for average repair time.Consider the small random subset of the Verizon data in Exercise 16.1.Bootstrap the sample mean using 1000 resamples.(a) Make a histogram and Normal quantile plot. Does the bootstrap distribution appear close to Normal? Is the
16.30 Determining the percentile endpoints. What percentiles of the bootstrap distribution are the endpoints of a 90% bootstrap percentile confidence interval? Of a 98% bootstrap percentile confidence interval?
16.29 The effect of non-Normality. The populations in the two previous exercises have the same mean and standard deviation, but one is very close to Normal and the other is strongly non-Normal. Based on your work in these exercises, how does non-Normality of the population affect the bootstrap
16.28 The effect of increasing sample size. The data for Example 16.1 are 1664 repair times for customers of Verizon, the local telephone company in their area. In that example, these observations formed a sample. Now we will treat these 1664 observations as a population. The population
16.27 Bootstrap versus sampling distribution. Most statistical software includes a function to generate samples from Normal distributions. Set the mean to 8.4 and the standard deviation to 14.7. You can think of all the numbers that would be produced by this function if it ran forever as a
16.26 Comparing the average repair time bootstrap distributions. Why is the bootstrap distribution of the difference in mean Verizon and CLEC repair times in Figure 16.9 so skewed? Let’s investigate by bootstrapping the mean of the CLEC data and comparing it with the bootstrap distribution for
16.25 CHALLENGE The really rich. Each year, the business magazine Forbes publishes a list of the world’s billionaires. In 2006, the magazine found 793 billionaires. Here is the wealth, as estimated by Forbes and rounded to the nearest $100 million, of an SRS of 20 of these billionaires:9 2.9 15.9
16.24 Bootstrap distribution of the mpg standard deviation. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the mpg were recorded
16.23 Do you feel lucky? Exercise 7.29 (page 443) gives data on 60 children who said how big a part they thought luck played in solving puzzles. The data have a discrete 1 to 10 scale. Is inference based on t distributions nonetheless justified? Explain your answer. If t inference is justified,
16.22 Bootstrap distribution of the median.We will see in Section 16.3 that bootstrap methods often work poorly for the median. To illustrate this, bootstrap the sample median of the 50 selling prices in Table 16.1. Why is the bootstrap t confidence interval not justified?
16.21 Bootstrapping a Normal data set. The following data are “really Normal.” They are an SRS from the standard Normal distribution N(0, 1), produced by a software Normal random number generator.0.01 −0.04 −1.02 −0.13 −0.36 −0.03 −1.88 0.34 −0.00 1.21−0.02 −1.01 0.58 0.92
16.20 Bootstrap comparison of tree diameters.In Exercise 7.81 (page 471) you were asked to compare the mean diameter at breast height(DBH) for trees from the northern and southern halves of a land tract using a random sample of 30 trees from each region.(a) Use a back-to-back stemplot or
16.19 Bootstrap distribution of the standard deviation s. For Example 16.5 we bootstrapped the 25%trimmed mean of the 50 selling prices in Table 16.1. Another statistic whose sampling distribution is unknown to us is the standard deviation s.Bootstrap s for these data. Discuss the shape and bias of
16.18 Another bootstrap distribution of the trimmed mean. Bootstrap distributions and quantities based on them differ randomly when we repeat the resampling process. A key fact is that they do not differ very much if we use a large number of resamples. Figure 16.7 shows one bootstrap distribution
16.17 Bootstrap t confidence interval for survival times. Return to or re-create the bootstrap distribution of the sample mean for the 72 guinea pig survival times in Exercise 16.10.(a) What is the bootstrap estimate of the bias?Verify from the graphs of the bootstrap distribution that the
16.16 Bootstrap t confidence interval for listening times. Return to or re-create the bootstrap distribution of the sample mean for the 8 listening times in Exercise 16.6.(a) Although the sample is small, verify using graphs and numerical summaries of the bootstrap distribution that the
16.15 Formula-based versus bootstrap standard error. We have a formula(page 450) for the standard error of x1 − x2. This formula does not depend on Normality. How does this formula-based standard error for the data of Example 16.6 compare with the bootstrap standard error?
16.14 Bootstrap comparison of average reading abilities. Table 7.4(page 452) gives the scores on a test of reading ability for two groups of third-grade students. The treatment group used “directed reading activities,” and the control group followed the same curriculum without the
16.13 Bootstrap t confidence interval for average audio file length. Return to or create the bootstrap distribution resamples on the sample mean for the audio file lengths in Exercise 16.8. In Example 7.11(page 436), the t confidence interval for the average length was constructed.(a) Inspect the
16.12 Bootstrap t confidence interval for repair times. Refer to Exercise 16.1. Suppose a bootstrap distribution was created using 1000 resamples, and the mean and standard deviation of the resample sample means were 13.762 and 4.725, respectively.(a) What is the bootstrap estimate of the bias?(b)
16.11 More on survival times in a medical study. Here is an SRS of 20 of the guinea pig survival times from Exercise 16.10:92 123 88 598 100 114 89 522 58 191 137 100 403 144 184 102 83 126 53 79 We expect the sampling distribution of x to be less close to Normal for samples of size 20 than for
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